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In the standard model of non-linear structure formation, a cosmic web of dark-matter dominated filaments connects dark matter halos. In this paper, we stack the weak lensing signal of an ensemble of filaments between groups and clusters of galaxies. Specifically, we detect the weak lensing signal, using CFHTLenS galaxy ellipticities, from stacked filaments between SDSS-III/BOSS luminous red galaxies (LRGs). As a control, we compare the physical LRG pairs with projected LRG pairs that are more widely separated in redshift space. We detect the excess filament mass density in the projected pairs at the $5\sigma$ level, finding a mass of $(1.6 \pm 0.3) \times 10^{13} M_{\odot}$ for a stacked filament region 7.1 $h^{-1}$ Mpc long and 2.5 $h^{-1}$ Mpc wide. This filament signal is compared with a model based on the three-point galaxy-galaxy-convergence correlation function, as developed in Clampitt, Jain & Takada (2014), yielding reasonable agreement.